The minimum restricted edge-connected graph and the minimum size of graphs with a given edge–degree

Let G=(V(G),E(G))G=(V(G),E(G)) be a graph. Determining the minimum and/or maximum size (|E(G)||E(G)|) of graphs with some given parameters is a classic extremal problem in graph theory. For a graph GG and e=uv∈E(G)e=uv∈E(G), we denote d(e)=d(u)+d(v)−2d(e)=d(u)+d(v)−2 the edge–degree of ee. In this paper, we obtain a lower bound for the minimum size of graphs with a given order nn, a given minimum degree δδ and a given minimum edge–degree 2δ+k−22δ+k−2. Moreover, we characterize the extremal graphs for k=0,1,2k=0,1,2. As an application, we characterize some kinds of minimum restricted edge connected graphs.